The Buchweitz Set of a Numerical Semigroup

Abstract

Let A subset mathbb Z be a finite subset. We denote by {{,mathrm{mathcal {B}},}}(A) the set of all integers n ge 2 such that |nA| > (2n-1)(|A|-1), where nA=A+cdots +A denotes the n-fold sumset of A. The motivation to consider {{,mathrm{mathcal {B}},}}(A) stems from Buchweitz’s discovery in 1980 that if a numerical semigroup S subseteq mathbb N is a Weierstrass semigroup, then {{,mathrm{mathcal {B}},}}(mathbb N{setminus } S) = emptyset . By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (Math Ann 41:403–442, 1893). In this paper, we prove that for any numerical semigroup S subset mathbb N of genus g ge 2, the set {{,mathrm{mathcal {B}},}}(mathbb N{setminus } S) is finite, of unbounded cardinality as S varies.